Search

The Newtonian potential of an Euclidean ball $B$ of $\mathbb{R}^n$ centered at $x_0$ is proportional, outside $B$, to the Newtonian potential of a mass concentrated at $x_0$. Vice-versa, as proved by Aharonov, Schiffer and Zalcman, if $D$ is a bounded open set in $\mathbb{R}^n$, containing $x_0$, whose Newtonian potential is proportional, outside $D$, to the one of a mass concentrated at $x_0$, then $D$ is an Euclidean ball with center $x_0$. In this paper we generalize this last result to more general measures and domains.

Let H be a linear second order partial differential operator with non-negative characteristic form in a strip S ⊂ R^N ×R. We assume that H as a fundamental solution, smooth out of its poles and bounded from above and from below by Gaussian kernels modeled on subriemannian doubling distances in R^N. Under these assumptions we show that H endows S with a structure of β-harmonic space. This allows us to study boundary value problems for L with a Perron-Wiener-Brelot-Bauer method, and to obtain pointwise regularity estimates at the boundary in terms of Wiener series modeled on the Gaussian kernels. Our analysis includes the proof of a scale invariant Harnack inequality for nonnegative solutions. We also show an application to the real hypersurphaces of C^{n+1} with given Levi-curvature.

We establish Holder estimates of second derivatives for a class of sub-elliptic partial differential operators in \({\mathbb{R}^{N}}\) of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the Xj’s are smooth vector fields in \({\mathbb{R}^{N}}\), and aij is a uniformly elliptic matrix. It is assumed that the Xj’s satisfy homogeneity conditions with respect to a group of dilations δr which yield the existence of a composition law \({\circ}\) in \({\mathbb{R}^{N}}\) making the triplet \({\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}\) an homogeneous Lie group on which the Xj’s are left translation invariant. The Holder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on \({\mathbb{G}}\), some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderon and Zygmund in the Euclidean setting.

reserved 3 no In this paper, we are concerned with hypoelliptic diffusion operators H. Our main aim is to show, with an axiomatic approach, that a Wiener-type test of H-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of H with respect to a distance satisfying doubling condition and segment property. As a main step toward this result, we establish some estimates at the boundary of the continuity modulus for the generalized Perron–Wiener solution to the relevant Dirichlet problem. The estimates involve Wiener-type series, with the capacities modeled on the Gaussian bounds. We finally prove boundary Hölder estimates of the solution under a suitable exterior cone condition. mixed Lanconelli, Ermanno; Tralli, Giulio; Uguzzoni, Francesco Lanconelli, Ermanno; Tralli, Giulio; Uguzzoni, Francesco

The aim of this paper is to give some representation formulas of Riesz and Poisson-Jensen type for super-solutions to a class of hypoelliptic ultraparabolic operators \(\mathcal{L}\) on a homogeneous Lie group \(\mathbb{L}\). Our results complete the ones obtained in Cinti (Math Scand 100:1–21, 2007). We also provide a suitable theory for \(\mathcal{L}\)-Green functions and for \(\mathcal{L}\)-Green potentials of Radon measures. The proofs mostly rely on the use of appropriate techniques relevant to the Potential Theory for \(\mathcal{L}\).

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bocher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).

We prove the existence of a fundamental solution for a class of Hormander heat-type operators. For this fundamental solution and its derivatives we obtain sharp Gaussian bounds that allow to prove an invariant Harnack inequality. To cite this article: M. Bramanti et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

We construct the fundamental solutionsΓ\Gammaandγ\gammafor the non-divergence form operators∑i,jai,j(x,t)XiXj−∂t\,{\textstyle \sum _{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial _t\,and∑i,jai,j(x)XiXj{\,\textstyle \sum _{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j, where theXiX_i’s are Hörmander vector fields generating a stratified groupG\mathbb {G}and(ai,j)i,j(a_{i,j})_{i,j}is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates ofΓ\Gammaand its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates ofΓ\Gammaallow us to constructγ\gammausing bothtt-saturation and approximation arguments.

Many results of classical Potential Theory are extended to sub-Laplacians ▵𝔾 on Carnot groups 𝔾. Some characterizations of ▵𝔾-subharmonicity, representation formulas of Poisson-Jensen's kind and Nevanlinna-type theorems are proved. We also characterize the Riesz-measure related to bounded-above ▵𝔾-subharmonic functions in ℝ N .

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.

A characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x → Ax of the Euclidean case must be replaced by suitable group isomorphisms x → T_A(x), whose differential preserves the first layer of the stratification of Lie(G).

We introduce a new class of curvature PDO's describing relevant properties of real hypersurfaces of $C^{n+1}.$ In our setting the pseudoconvexity and the Levi form play the same role as the convexity and the real Hessian matrix play in the real Euclidean one. Our curvature operators are second order fully nonlinear PDO's not elliptic at any point. However, when computed on generalized $s$-pseudoconvex functions, we shall show that their characteristic form is nonnegative definite with kernel of dimension one. Moreover, we shall show that the missing ellipticity direction can be recovered by taking into account the CR structure of the hypersurfaces. These properties allow us to prove a strong comparison principle, leading to symmetry theorems for domains with constant curvatures and to identification results for domains with comparable curvatures.